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REVIEW 4 major objections 3 minor 45 references

DDRM-PR: Fourier Phase Retrieval using Denoising Diffusion Restoration Models

T0 review · 4 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Phase retrieval improves when a diffusion prior refines alternating-projection reconstructions.

desk verdict A genuinely new training-free combination of DDRM and HIO for phase retrieval, with an honest limitations section, but the central analogy that HIO approximates the pseudo-inverse is a load-bearing heuristic that the paper does not verify. read the letter →

arxiv 2501.03030 v1 pith:64ISTEK2 submitted 2025-01-06 eess.IV cs.CV

Mehmet Onurcan Kaya , Figen S. Oktem This is my paper · ORCID
classification eess.IVcs.CV
keywords phaseretrievalFouriermagnitudediffusionmodelsdenoisingrestorationhybridinput-outputalternatingprojectionsinverseproblemspretrainedpriors
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that Fourier phase retrieval—recovering an image from noisy measurements of only the magnitude of its Fourier transform—can be improved by wrapping the classical Hybrid Input-Output (HIO) alternating-projection algorithm inside a Denoising Diffusion Restoration Model (DDRM). The difficulty is that DDRM was derived for linear inverse problems; the paper's move is to let HIO play the role of the pseudo-inverse $H^{\dagger}$ in the noiseless DDRM update, producing a training-free update that uses a pretrained unconditional diffusion model as the image prior. On simulated face images the method reports higher PSNR and SSIM and lower LPIPS than HIO alone at every tested noise level, and on experimental imaging-through-scattering data it reports a large quantitative improvement. The paper also argues the same recipe extends to other phase retrieval settings, such as coded diffraction patterns and complex-valued images, provided a matching pretrained denoiser is available.

What carries the argument

The load-bearing device is the analogy between the Moore–Penrose pseudo-inverse $H^{\dagger}$ in the linear DDRM update and the HIO algorithm—an alternating-projection method that alternates between Fourier magnitude constraints and spatial constraints—applied to Fourier magnitudes. In the noise-free case, HIO is expected to return the original image from its Fourier magnitude, mirroring $H^{\dagger}H x \approx x$, and to leave the noiseless measurement unchanged, mirroring $H H^{\dagger}H = H$. The authors use this analogy to rewrite the correction term as $\text{HIO}(|F f_\theta^{(t+1)}(x_{t+1})|)$, with a random-initialization HIO step supplying the data-consistency component. Everything else in the pipeline—the pretrained denoiser, the noise schedule, and the averaging over multiple outputs—is standard DDRM machinery.

What would settle it

Take a noiseless natural image $x$, compute the HIO output from its Fourier magnitude with the paper's parameters, and compare both the reconstruction error and the Fourier-magnitude residual of that output; if for a nontrivial fraction of images the update $x - \text{HIO}(|Fx|)$ is not closer to $x$ than before, or the magnitude residual is not near zero, the data-consistency correction in Eq. (13) is not doing what the pseudo-inverse analogy claims.

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Extended reading notes

Core claim

The central claim is that the simplified noiseless DDRM update $x'_t = x_{\theta,t} - H^{\dagger}H x_{\theta,t} + H^{\dagger} y$ remains valid for the nonlinear Fourier phase retrieval operator if HIO, run on the Fourier magnitude of the denoiser output, is substituted for the pseudo-inverse terms. The resulting update is Eq. (13): $x'_t = f_\theta^{(t+1)}(x_{t+1}) - \text{HIO}(|F f_\theta^{(t+1)}(x_{t+1})|) + \text{RandomInit}(y)$, followed by the standard diffusion interpolation with noise. The paper treats this as a regularized extension of alternating projection: the diffusion prior cleans HIO artifacts, while the HIO term keeps the iterate consistent with the measured Fourier magnitude. It reports that this combination outperforms the HIO initialization on all evaluated metrics in simulations and substantially improves PSNR, SSIM, and LPIPS on experimental scattering data, with the caveat that HIO artifacts can persist and the noisy-measurement theory is not developed.

Load-bearing premise

The whole update hinges on the unproven assumption that HIO acts like the linear pseudo-inverse $H^{\dagger}$: that running HIO on the Fourier magnitude of a denoised image brings it closer to the true image and preserves the measured magnitude, rather than getting trapped in a local minimum with large residual error.

Editorial extensions

If this is right

  • The same recipe can be applied to any phase retrieval problem with an alternating-projection solver, so coded diffraction patterns, ptychography, and scattering measurements become candidates for diffusion-prior refinement without retraining.
  • Using a pretrained unconditional diffusion model means a new measurement setup does not require collecting paired training data; only the alternating-projection component needs to be matched to the measurement.
  • Simulation results imply that even when the baseline HIO reconstruction is noisy, the diffusion update improves both distortion metrics (PSNR, SSIM) and perceptual quality (LPIPS) across the tested noise range.
  • The experimental scattering results suggest the prior generalizes beyond its natural-image training distribution, since a black-and-white target is recovered with a model trained on color natural images.
  • Because the method still inherits some HIO artifacts, its ceiling is tied to the quality of the alternating-projection initialization; better initializations would translate directly into better final reconstructions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the HIO $\approx H^{\dagger}$ analogy is quantitatively tested on natural images, a large gap between HIO$(|Fx|)$ and $x$ would predict where Eq. (13) injects the wrong target; this is a testable diagnostic the paper does not report.
  • The method's per-step cost includes a 100-iteration HIO run inside every diffusion step, so timings on a standard GPU would clarify whether the gains are practical for real-time imaging; the paper does not report runtime.
  • The claimed extension to complex-valued images likely requires a pretrained diffusion model whose training distribution matches the amplitude and phase statistics of the target, which may be harder to obtain than the natural-image prior used here.
  • One could test the prior's role directly by replacing the pretrained diffusion denoiser with a classical denoiser of similar strength; if the gain mostly disappears, the diffusion sampling procedure rather than the prior is what matters.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The paper proposes DDRM-PR, a training-free method for Fourier phase retrieval that combines the simplified noiseless DDRM update with the classical HIO alternating-projection algorithm as a stand-in for the pseudo-inverse of the nonlinear Fourier-magnitude operator. The method uses a pretrained unconditional diffusion model to refine HIO reconstructions, with a random-init procedure and averaging over multiple samples. Experiments on simulated CelebA-HQ data and on experimental scattering-medium data report consistent improvements over HIO in PSNR, SSIM, and LPIPS. The authors also provide a proof of the simplified linear DDRM update in an appendix and discuss extensions to other phase retrieval settings.

Significance. If the central heuristic were rigorously validated, DDRM-PR would be a useful addition to the phase retrieval toolbox, demonstrating a plug-and-play use of pretrained diffusion priors for a nonlinear inverse problem without task-specific training. The paper is commendable for providing source code, testing on experimental transmission-matrix data, and being transparent about the heuristic nature of the HIO substitution and about the method's remaining artifacts. However, the significance currently rests on an unproven analogy between HIO and the linear pseudo-inverse, and the empirical evaluation is too thin to establish that the proposed mechanism, rather than the unconditional diffusion prior or sample averaging, is responsible for the reported gains.

major comments (4)
  1. [§4.1, Eq. (13)] The load-bearing assumption that HIO can replace the pseudo-inverse H† in the simplified DDRM update is not justified. The two bullet-point observations about the noiseless case establish only that HIO, when initialized appropriately and run to convergence, can recover the image up to trivial ambiguities and reproduce the Fourier magnitude; they do not imply that the concrete operator used in Eq. (13), namely HIO(|F fθ(x_{t+1})|) initialized from x_{t+1} and run for k=100 iterations, behaves like the orthogonal projection H†H along the entire diffusion trajectory, nor that it preserves data consistency under the noisy measurements used in the experiments. The paper provides no direct evidence for these properties and no ablation that removes the HIO correction or replaces it with an exact projection, so the reader cannot tell whether the improvement over HIO arises from the claimed DDRM mechanism or from the diffusion prior and averaging alone.
  2. [Appendix A, Eq. (32)] The derivation of the simplified DDRM form uses the approximation sqrt(1-η²) ≈ 1-η without any error bound or justification. For intermediate values such as η=0.5, the two expressions differ by about 0.37 in absolute value, and this error is multiplied by the noise term in Eq. (32) and propagated into the final update. Since Theorem 4.1 is stated as a formal result, the approximation should either be removed by an exact derivation or stated as an explicit approximation whose validity range is quantified; as written, the proof does not establish the claimed equality.
  3. [§5.1 and §5.2] The evaluation protocol is insufficiently specified for the reported numbers to be fully credible. The grid search over η, η_b, t, and T_init is described as 'linear grid search' but no validation split is mentioned; if the hyperparameters were selected on the test set, the reported PSNR/SSIM/LPIPS values are optimistic. In addition, Table 1 reports averages without error bars or the number of test images, and the experimental section reports a single configuration. The paper should include a clear train/validation/test split for hyperparameter selection, report variability across the test set, and state the number of images used for each entry.
  4. [§5.1, Table 1] The comparison is limited to a single baseline (HIO), and that baseline is also the initialization stage of the proposed method. This makes it impossible to attribute the improvement to the diffusion prior, the HIO correction, the random-init procedure, or the averaging over N=8 samples. At minimum, the authors should include an ablation with the diffusion prior but without the HIO correction term, an ablation with the HIO correction but without the diffusion prior, and a comparison with a stronger classical or learning-based PR baseline such as prDeep or oversampling smoothness; without such controls the central claim of 'improving alternating-projection methods' is not convincingly supported.
minor comments (3)
  1. [§4.1] The sentence 'RandomInitrepresentstotheHIOinitializationprocedureproposedintheprDeeppaper' contains a typographical error ('representstothe') and would benefit from rewording for clarity.
  2. [Figures 1–4] The figure captions list hyperparameter values such as η=0.15 and t=15, but the main text states that a grid search was performed; it is unclear whether the values shown in the captions are the selected optimal values for each noise level or just illustrative examples. Please clarify.
  3. [Appendix A, Definition A.2] The use of cyan color to denote scaled variables in Definition A.2 is not visible in a black-and-white print version and is not self-explanatory; please use an explicit notation such as a tilde or superscript instead of or in addition to color.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: DDRM-PR is a heuristic substitution of HIO for the pseudo-inverse in DDRM, supported by external empirical comparison rather than by a fit or self-citation.

full rationale

The derivation of the simplified DDRM update (Eq. 12) is self-contained in Appendix A, starting from the published DDRM sampler [17] and the noiseless form [19]; the proof is algebraic and does not assume the phase-retrieval result. The new update in Eq. (13) is obtained by explicitly replacing H†H and H†y with HIO and RandomInit, with the two bullet-point justifications stated as observations about the noiseless case. This is an unverified analogy (HIO is nonlinear and initialization-dependent), which is a correctness risk, but it is not circular: HIO is not defined in terms of DDRM-PR's output, and no fitted parameter is hidden in the update. The later performance claims are empirical comparisons against HIO, not consequences of the equations. Self-citations (e.g., refs. 8, 13, 14, 44) concern earlier phase-retrieval work and are not used as the load-bearing justification for the HIO-as-pseudo-inverse step, which is offered as a heuristic. The grid-search choice of η, η_b, t, T_init, N is hyperparameter tuning, a separate methodological concern, not a circular derivation.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The method rests on the Gaussian measurement model, the uniqueness of oversampled Fourier magnitudes, the heuristic identification of HIO with a pseudo-inverse, an approximation in the proof of the simplified DDRM update, and the assumption that a natural-image diffusion model is a suitable prior for phase retrieval targets. Hyperparameters (η, η_b, t, T_init, N) are chosen by grid search and are best treated as fitted values. No new physical entities are introduced.

free parameters (6)
  • eta (η) = 0.15 (α=0.5), 0.25 (α=1), 0.25 (α=2), 0.78 (α=3), 1.0 (experimental)
    DDRM transition variance hyperparameter selected by grid search per noise level; affects the update step in Eq. (13).
  • eta_b (η_b) = 0.20 (α=0.5), 0.22 (α=1), 0.18 (α=2), 0.17 (α=3), 0.0 (experimental)
    DDRM data-consistency mixing hyperparameter selected by grid search per noise level.
  • diffusion timestep t = 15, 30, 15, 30, 35 across settings
    Number of DDRM refinement steps chosen by grid search; affects reconstruction quality.
  • initial timestep T_init = 350, 400, 400, 300, 220 across settings
    Starting diffusion timestep for sampling, chosen by grid search.
  • number of averaged reconstructions N = 8 in Section 5.1 text; 1 in figure captions
    Outputs are averaged for stability; inconsistent reporting between text and captions.
  • HIO iterations and random initializations = k=100, s=50, n=1000
    HIO inner-loop iterations and RandomInit runs; chosen by the authors, not swept.
assumptions (5)
  • domain assumption Measurement noise in Eq. (1) is Gaussian with variance proportional to the true Fourier intensity.
    Assumed to approximate Poisson noise; stated in Section 2. The method further ignores this noise in the simplified noiseless DDRM update.
  • standard math Oversampled Fourier magnitudes uniquely determine a real, nonnegative, finite-support image (m=4n).
    Cited from Hayes [20]; underpins the phase retrieval well-posedness.
  • ad hoc to paper HIO acts as a pseudo-inverse for the nonlinear Fourier magnitude operator, analogously to H†H in linear DDRM.
    Core heuristic in Section 4.1; not proven, and shown to fail when HIO is trapped in local minima.
  • ad hoc to paper The approximation sqrt(1-η^2) ≈ 1-η used in the proof of Theorem 4.1.
    Appendix A, after Eq. (31); exact only at η=0 or η=1, not for interior values used in experiments.
  • domain assumption A pretrained unconditional diffusion model trained on natural images approximates the target image distribution for phase retrieval.
    Section 4.1 and used in experiments; the experimental scattering targets are grayscale, mismatched with ImageNet training data.

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Cite this review

Pith. "Pith review of DDRM-PR: Fourier Phase Retrieval using Denoising Diffusion Restoration Models." pith.science (2026). https://pith.science/paper/64ISTEK2

@misc{pith2026250103030,
  author       = {Pith},
  title        = {Pith review of: DDRM-PR: Fourier Phase Retrieval using Denoising Diffusion Restoration Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/64ISTEK2}},
  note         = {Machine review of arXiv:2501.03030}
}
read the original abstract

Diffusion models have demonstrated their utility as learned priors for solving various inverse problems. However, most existing approaches are limited to linear inverse problems. This paper exploits the efficient and unsupervised posterior sampling framework of Denoising Diffusion Restoration Models (DDRM) for the solution of nonlinear phase retrieval problem, which requires reconstructing an image from its noisy intensity-only measurements such as Fourier intensity. The approach combines the model-based alternating-projection methods with the DDRM to utilize pretrained unconditional diffusion priors for phase retrieval. The performance is demonstrated through both simulations and experimental data. Results demonstrate the potential of this approach for improving the alternating-projection methods as well as its limitations.

Figures

Figures reproduced from arXiv: 2501.03030 by the authors.

Figure 1
Figure 1. Ground-truth test images (top row), reconstructions using the developed [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Ground-truth test images (top row), reconstructions using the developed [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Ground-truth test images (top row), reconstructions using the developed [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Ground-truth test images (top row), reconstructions using the developed [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Performance comparison with experimental data. Each row corresponds [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

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    2024

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.